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(d^2u)/(dz^2)+(du)/(dz)+(k/z+(1/4-m^2)/(z^2))u=0. (1) Let u=e^(-z/2)W_(k,m)(z), where W_(k,m)(z) denotes a Whittaker function. Then (1) becomes ...

A partial differential equation of second-order, i.e., one of the form Au_(xx)+2Bu_(xy)+Cu_(yy)+Du_x+Eu_y+F=0, (1) is called parabolic if the matrix Z=[A B; B C] (2) ...

The second-order ordinary differential equation y^('')+[A+Bcos(2x)+Ccos(4x)]y=0.

An elliptic partial differential equation given by del ^2psi+k^2psi=0, (1) where psi is a scalar function and del ^2 is the scalar Laplacian, or del ^2F+k^2F=0, (2) where F ...

The second-order ordinary differential equation y^('')+alpha(x)y^'+x^2y^n=0.

Whittaker and Watson (1990, pp. 539-540) write Lamé's differential equation for ellipsoidal harmonics of the first kind of the four types as ...

The third-order ordinary differential equation y^(''')+alphayy^('')+beta(1-y^('2))=0.

The Helmholtz differential equation is not separable in bispherical coordinates.

The Helmholtz differential equation is not separable in toroidal coordinates

The second-order ordinary differential equation y^('')-[(m(m+1)+1/4-(m+1/2)cosx)/(sin^2x)+(lambda+1/2)]y=0.